On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs

Abstract

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in (formula presented) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no (formula presented)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Expo-nential Time Hypothesis, for any fixed q, q-Colouring does not admit a (formula presented)-time algorithm, even when restricted to unit disk graphs, and it is solvable in (formula presented)-time on disk graphs.